Solution 23 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
2 2
35*a20 *a4 *b36
b73=-----------------
4
a13
35 2 2
----*a20 *a4 *b36
8
b72=-------------------
4
a13
b71=0
3
80*a20*a4 *b36
b70=----------------
4
a13
3
25*a20*a4 *b36
b69=----------------
4
a13
3
70*a20*a4 *b36
b68=----------------
4
a13
b67=0
b66=0
b65=0
b64=0
2
80*a20*a4 *b36
b63=----------------
3
a13
2
25*a20*a4 *b36
b62=----------------
3
a13
2
35*a20*a4 *b36
b61=----------------
3
a13
- 15*a20*a4*b36
b60=------------------
2
a13
- 35*a20*a4*b36
b59=------------------
2
a13
- 20*a20*a4*b36
b58=------------------
2
a13
35
- ----*a20*a4*b36
2
b57=--------------------
2
a13
- 15*a20*a4*b36
b56=------------------
2
a13
5
- ---*a20*a4*b36
2
b55=-------------------
2
a13
b54=0
4
30*a4 *b36
b53=------------
4
a13
4
120*a4 *b36
b52=-------------
4
a13
b51=0
3
60*a4 *b36
b50=------------
3
a13
3
120*a4 *b36
b49=-------------
3
a13
3
60*a4 *b36
b48=------------
3
a13
2
- 10*a4 *b36
b47=---------------
2
a13
2
- 40*a4 *b36
b46=---------------
2
a13
2
- 10*a4 *b36
b45=---------------
2
a13
2
- 40*a4 *b36
b44=---------------
2
a13
b43=0
2
- 20*a4 *b36
b42=---------------
2
a13
b41=0
- 10*a4*b36
b40=--------------
a13
- 20*a4*b36
b39=--------------
a13
- 20*a4*b36
b38=--------------
a13
- 10*a4*b36
b37=--------------
a13
b35=0
b34=0
35 2 2
----*a20 *a4 *b36
2
b33=-------------------
4
a13
35 2 2
----*a20 *a4 *b36
8
b32=-------------------
4
a13
35 2
- ----*a20 *a4*b36
2
b31=---------------------
3
a13
35 2
- ----*a20 *a4*b36
4
b30=---------------------
3
a13
35 2
- ----*a20 *a4*b36
4
b29=---------------------
3
a13
b28=0
3
30*a20*a4 *b36
b27=----------------
4
a13
3
45*a20*a4 *b36
b26=----------------
4
a13
2
- 15*a20*a4 *b36
b25=-------------------
3
a13
2
- 15*a20*a4 *b36
b24=-------------------
3
a13
2
- 30*a20*a4 *b36
b23=-------------------
3
a13
b22=0
b21=0
- 5*a20*a4*b36
b20=-----------------
2
a13
- 10*a20*a4*b36
b19=------------------
2
a13
- 20*a20*a4*b36
b18=------------------
2
a13
35
- ----*a20*a4*b36
2
b17=--------------------
2
a13
- 30*a20*a4*b36
b16=------------------
2
a13
15
- ----*a20*a4*b36
2
b15=--------------------
2
a13
5
---*a20*b36
2
b14=-------------
a13
15
----*a20*b36
2
b13=--------------
a13
5*a20*b36
b12=-----------
a13
b11=0
4
30*a4 *b36
b10=------------
4
a13
b9=0
b8=0
2
- 10*a4 *b36
b7=---------------
2
a13
2
- 40*a4 *b36
b6=---------------
2
a13
2
- 10*a4 *b36
b5=---------------
2
a13
b4=0
b3=0
b2=0
b1=b36
a21=0
1
a19=---*a20
4
a18=0
a17=a4
a16=2*a4
a15=0
a14=a13
1 2
- ---*a13
6
a12=-------------
a4
a11=0
a10=0
1
a9=---*a20
2
1
a8=---*a20
4
1
- ---*a13*a20
4
a7=----------------
a4
1
- ---*a13*a20
4
a6=----------------
a4
a5=0
a3=0
a2=0
1 2
- ---*a13
6
a1=-------------
a4
Parameters
Apart from the condition that they must not vanish to give
a non-trivial solution and a non-singular solution with
non-vanishing denominators, the following parameters are free:
b36,a20,a13,a4
Relevance for the application:
The solution given above tells us that the system {u_s, v_s}
is a higher order symmetry for the lower order system {u_t,v_t}
where u=u(t,x) is a scalar function, v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1 2 1 2 2 1
u =( - ---*u *a13 + ---*u *f(v,v)*a20*a4 + u *a4 *u - ---*f(v ,v )*a13*a20
t 6 3x 4 x x 4 x x
1 1
- ---*f(v,v )*a13*a20 + ---*f(v,v )*a20*a4*u)/a4
4 2x 2 x
2 1 2
v =(u *a13*a4*v + u *v *a13*a4 + 2*u *a4 *u*v - ---*v *a13
t 2x x x x 6 3x
1 2 2
+ ---*v *f(v,v)*a20*a4 + v *a4 *u + f(v,v )*a20*a4*v)/a4
4 x x x
4 15 2 2 2 2
u =(u *a13 *b36 - ----*u *f(v,v)*a13 *a20*a4*b36 - 10*u *a13 *a4 *b36*u
s 5x 2 3x 3x
2 2 2
- 40*u *u *a13 *a4 *b36*u - 30*u *f(v,v )*a13 *a20*a4*b36
2x x 2x x
3 2 2 35 2
- 10*u *a13 *a4 *b36 - ----*u *f(v ,v )*a13 *a20*a4*b36
x 2 x x x
2 2
- 20*u *f(v,v )*a13 *a20*a4*b36 - 30*u *f(v,v )*a13*a20*a4 *b36*u
x 2x x x
35 2 2 2 3 2
+ ----*u *f(v,v) *a20 *a4 *b36 + 45*u *f(v,v)*a20*a4 *b36*u
8 x x
4 4 3
+ 30*u *a4 *b36*u + 5*f(v ,v )*a13 *a20*b36
x 2x 2x
15 3 2
+ ----*f(v ,v )*a13 *a20*b36 - 10*f(v ,v )*a13 *a20*a4*b36*u
2 x 3x x 2x
35 2 2 2
- ----*f(v ,v )*f(v,v)*a13*a20 *a4*b36 - 15*f(v ,v )*a13*a20*a4 *b36*u
4 x x x x
5 3 2
+ ---*f(v,v )*a13 *a20*b36 - 5*f(v,v )*a13 *a20*a4*b36*u
2 4x 3x
35 2 2 2
- ----*f(v,v )*f(v,v)*a13*a20 *a4*b36 - 15*f(v,v )*a13*a20*a4 *b36*u
4 2x 2x
35 2 2 35 2 2
- ----*f(v,v ) *a13*a20 *a4*b36 + ----*f(v,v )*f(v,v)*a20 *a4 *b36*u
2 x 2 x
3 3 4
+ 30*f(v,v )*a20*a4 *b36*u )/a13
x
3 3 2 2
v =( - 10*u *a13 *a4*b36*v - 20*u *v *a13 *a4*b36 - 20*u *a13 *a4 *b36*u*v
s 4x 3x x 3x
3 2 2
- 20*u *v *a13 *a4*b36 - 40*u *v *a13 *a4 *b36*u
2x 2x 2x x
2 3 2
+ 35*u *f(v,v)*a13*a20*a4 *b36*v + 60*u *a13*a4 *b36*u *v
2x 2x
2 2 2 2 3 3
- 10*u *v *a13 *a4 *b36 + 120*u *a13*a4 *b36*u*v - 10*u *v *a13 *a4*b36
x x x x 3x
2 2 2
- 40*u *v *a13 *a4 *b36*u + 25*u *v *f(v,v)*a13*a20*a4 *b36
x 2x x x
3 2 2
+ 60*u *v *a13*a4 *b36*u + 80*u *f(v,v )*a13*a20*a4 *b36*v
x x x x
3 4 3 4
+ 70*u *f(v,v)*a20*a4 *b36*u*v + 120*u *a4 *b36*u *v + v *a13 *b36
x x 5x
5 2 2 2 2
- ---*v *f(v,v)*a13 *a20*a4*b36 - 10*v *a13 *a4 *b36*u
2 3x 3x
2 35 2
- 15*v *f(v,v )*a13 *a20*a4*b36 - ----*v *f(v ,v )*a13 *a20*a4*b36
2x x 2 x x x
2 35 2 2 2
- 20*v *f(v,v )*a13 *a20*a4*b36 + ----*v *f(v,v) *a20 *a4 *b36
x 2x 8 x
3 2 4 4
+ 25*v *f(v,v)*a20*a4 *b36*u + 30*v *a4 *b36*u
x x
2 2
- 35*f(v ,v )*a13 *a20*a4*b36*v - 15*f(v,v )*a13 *a20*a4*b36*v
x 2x 3x
2 2 3 2 4
+ 35*f(v,v )*f(v,v)*a20 *a4 *b36*v + 80*f(v,v )*a20*a4 *b36*u *v)/a13
x x